Question

(ii) If A and B are the square matrices of the same order 3 , such that IAI =2 and AB = 2I where I is the identity matrix , write the value of IBI

Answer 1

Answer:

As we have learned

Property of Transpose -

\left ( A+B \right ){}'= A{}'+B{}'

- wherein

A and B being conformable matrices and A{}' is transpose of A

We have

A + B = 2B' .............(1)

A' + B' = 2B ...........(2)

Also 3A +2B = I_3

3A' + 2B' = I_3....(3)

from 1 and 3 \rightarrow A + B = I_3-3A'..........(4)

from 2 and 3 \rightarrow A' = I_3- 4B ...........(5)

from 4 and 5 \rightarrow A+B= I_3-3 (I_3-4B)

\Rightarrow A+B=-2I_3-12B

\Rightarrow11B-A = 2I_{3}..............(6)

now we have ,

3A+2B = I_3

11B -A = 2I_3

On solving ,

35B = 7 I_3

B = I_3/5

and

A = I_3 /5

10A + 5B = 3I3